2 Answers
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This article explains how the chief Rabbi is chosen and what are his powers.

Broadly speaking, the job combines two different roles. One is
ambassador of Judaism to the outside world; the other is "rabbi of
rabbis", the senior rabbi of the United Synagogue and other central
Orthodox communities. While the roles are not mutually exclusive, the
question is which will be given greater emphasis.

In theory, constitutionally, the chief rabbi is the supreme authority
of the United Synagogue, and the Beth Din is there to advise him. In
practice, it is the Beth Din that has been seen increasingly to call
the religious shots, and the current Chief Rabbi has been accused of
yielding too much ground to it.

His preference was to appoint a small working group of around seven
people who would draw up a shortlist and interview the candidates.
Their recommendation would be then submitted for approval to a larger
representative group of around 30, made up of delegates from various
constituencies under the aegis of the chief rabbi, but also possibly
from outside. Mr Pack proposes to chair both groups himself.

When Lord Sacks was appointed, the decision was taken by a body known
as the Chief Rabbinate Council, composed of around 200 members
although a smaller selection committee of 35 was set up to recommend a
name.

The council was superseded in 2002 by the much smaller Chief Rabbinate
Trust, comprising three United Synagogue officers, three other
trustees nominated by the United Synagogue and three trustees
representing regional and other communities.

I believe the previous chief rabbi, Lord Jacobovits, described the difference between his position and the Israeli chief rabbinate as: "My job has influence but no power; his has power but no influence."
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ShalomFeb 24 '12 at 0:10